Download Fluid Instabilities

Introduction
to
Fluid Instabilities
K. Yavuz Ekşi
What is Fluid Stability?
• A steady state solution of the
hydrodynamic equations
corresponding to a fluid configuration
may not exist or may not prevail if it
ever existed.
• There are always small perturbations
in any system.
• If such small perturbations grow in
time then the steady state solution is
unstable and can not serve as a
model for some realistic situation.
How are fluid instabilities relevant to
astrophysical phenomena?
• What are the solar granules?
• How does stars form out of gas clouds?
• How does galaxies form in an expanding
universe?
• Why are there spiral arms of galaxies?
• Turbulence in accretion disks: How is the angular
momentum transported outwards?
• Why do Dwarf Novae burst?
• The Great Red Spot of Jupiter...
• The clumpiness of matter in SNRs
Stability
• Linearize the equations around the equilibrium
• Check whether small perturbations grow.
linearly stable
non-linearly unstable
linearly stable
linearly unstable
Linearly unstable
Non-linearly stable
Gravitational Instability
• Jeans instability is the basic reason why
the matter in the Universe is not spread
uniformly
• Stars and galaxies are believed to be the
end products of perturbations which
initially started growing due to the Jeans
instability
Gravitational Instability: Jeans Criterion
• When a region of a gas is compressed, the excess
pressure there tries to smoothen out the compression:
 acoustic waves.
• The compressed region has enhanced gravitation
 more gas is pulled into the compressed region.
• For typical sound waves in the air, the enhancement of
gravity in the regions of compression is utterly negligible.
• For perturbations of gaseous bodies of astronomical size,
the enhanced gravitation in the region of compression
may overpower the expansive tendency of the excess
pressure,  more material pulled there, an instability
triggered.
Linear Stability Analysis for GI
Dynamical equations
Equilibrium Equations
Perturb the
Equilibrium
-Insert into the dynamical
equations .Neglect the 2nd
order terms
.Subtract the equilibrium
equations
Assume that all the perturbations are of the
form
Using these in the equations above
If k<kj then ! is
imaginary
and the perturbation
grows
exponentially…
where
Defining
And the enhanced
gravitation
in the region of
compression
may overpower the
expansive
tendency of the excess
pressure
Jeans mass
k<kj is equivalent to M>Mj
Assuming the interstellar matter to have 1
hydrogen atom per cm3 at temperature
100K, we obtain a Jeans mass of about
8×1038 g.
This is several orders of magnitude larger than the typical
mass of a star (about 1033 g)
Presumably the interstellar matter first breaks into large chunks with masses
corresponding to clusters of stars rather than individual stars.
Then somehow these contracting chunks of gas have to break further to
produce stars.
The presence of angular momentum or magnetic field complicates the
process.
Rayleigh-Taylor Instability (RTI)
• also called RichtmyerMeshkov instability,
• shows the competition
between surface
tension and gravity.
• Occurs anytime a
dense fluid is
accelerated by a light
fluid e.g. a heavy fluid
over a light fluid.
Visualisations of RTI
http://www.itsc.com/movies/raytay.mpg
2D fingers in Rayleigh-Taylor instability
When the RTI starts, the boundary between the fingers
and the surrounding medium is subject to KHI
3D fingers in Rayleigh-Taylor instability
Crab Nebula
The heavy fluid on top of light
fluid, by the principle of equivalence,
is same as heavy fluid accelerated
by a light fluid.
As the interface is decelerated,
we can represent it by an outward
gravitational field in the rest frame
of the interface. Then we have the
dense shell of ejected gas lying
on top of the less dense gas
outside. It was found by Gull
(1975) that this gives rise to the
Rayleigh-Taylor instability and
gives rise to the clumpy
appearance of material in Crab.
Simulations…
Fryxell, Müller & Arnett (1991) ApJ, 367, 619
Kelvin-Helmholtz Instability (KHI)
Kelvin-Helmholtz (inviscid)
instablility:
z
U 2 , 2
U1 , 1
x
linear stability analysis+normal mode disturbance:
k   U1  U 2    12  22  g k x2  k y2
2
x 1 2
2
KHI…
http://www.maths.man.ac.uk/~mheil/MATTHIAS/Fluid-Animations/kelvin.mpg
http://www.itsc.com/movies/kelvin.mpg
KHI in a jet
http://www.maths.man.ac.uk/~mheil/MATTHIAS/Fluid-Animations/bouyjet.mpg
http://www.itsc.com/movies/bouyjet.mpg
Kelvin Helmholtz
instablility
A long rectangular tube initially horizontal is filled with water above
colored brine The fluids are allowed to diffuse for about an hour, and
the tube then quickly tilted six degrees, setting the fluids into motion.
The brine accelerates uniformly down the slope, while the water
above similarly accelerates up the slope. Sinusoidal instability of the
interface occurs after a few seconds, and has here grown
nonlinearly into regular spiral rolls.
U 2 , 2
Kelvin-Helmholtz (inviscid)
instablility:
k   U1  U 2    12  22  g k x2  k y2
2
x 1 2
2
U1 , 1
~ instability due to heavy fluid on the upside
~ instability due to shear
~ instability due to an rapid downward vertical
acceleration
and heavy fluid rests below
~ instability for all cases
Wall Shear Flows
~ inviscidly unconditionally stable (Rayleigh
analysis)
~ viscously unstable (Orr-Sommerfeld analysis)
Re 
UD
 5772

~ unstable in labs as Re > 2000
Kelvin-Helmholtz as seen in a
cloud formation
From the National Center for Atmospheric Research.
http://venus.eng.deu.edu.tr/isilab/kelvin_helm_rollup.htm
Hiroshige Utagawa "Vortices in the
Konaruto stream"
In rivers, these structures can
appear when there is a
sudden widening of the river
bed.
Vincent Van Gogh "La Nuit Etoilee=Starry Night"
V838 Monocerotis
Expanding halo of light
around a distant star.
The illumination of interstellar
dust comes from the red
supergiant star at the
middle of the image, which
gave off a flashbulb-like pulse
of light two years before the
image was taken.
A KHI on Saturn, formed at the interaction of
two bands of the planet's atmosphere
Image taken from the Cassini probe of NASA
Transport of solar wind into Earth's magnetosphere
through rolled-up Kelvin−Helmholtz vortices
•
•
•
•
•
Nature 430, 755 (12 August 2004)
Hasegawa et al.
Establishing the mechanisms by
which the solar wind enters Earth's
magnetosphere is one of the biggest
goals of magnetospheric physics, as
it forms the basis of space weather
phenomena such as magnetic storms
and aurorae.
It is generally believed that magnetic
reconnection is the dominant process
However the plasma content in the
outer magnetosphere increases
during northward solar-wind magnetic
field conditions, contrary to
expectation if reconnection is
dominant
Here the authors show that during
northward solar-wind magnetic field
conditions—in the absence of active
reconnection at low latitudes—there
is a solar-wind transport mechanism
associated with the nonlinear phase
of the Kelvin−Helmholtz instability
Convective Instability
Sometimes we have to deal with
fluids heated from below.
Such flows have cold gases
overlying hot.
e.g. Atmosphere of the Earth.
When does such an adverse
temperature gradient become
unstable WRT the tendency to
develope overturning motions?
Schwarzschild
Criterion (1906) for
Convective instability:
ds>0 in the direction of gravity
Rayleigh-Bénard convection
• Convection cells in a
fluid is heated from
below.
Transition to convection of an initially quiescent layer of
fluid that has a vertically unstable temperature gradient.
http://www.itsc.com/movies/benard1.mpg
• Granules on the
solar surface are
the tops of
convection cells.
Thermal Instability
• Dwarf Novae: Opacity increases with
temperature => disk can not
cool=>becomes even hotter
Rotational Instability
• Inviscid case: Rayleigh Criterion (1917)
• Viscous case: Taylor Criterion (1923)
• Magnetic case: Velikhov
(1959)/Chandrasekhar(1961)
Couette Flow: a fluid confined
between two cylinders which
rotate at different rates. Results
apply to accretion disks.
Rotational Instability-Rayleigh criterion
Rayleigh Criterion for Instability
Specific angular momentum, j = r2, must decrease outwards.
• Accretion disks (=K=(GM/r3)1/2) can not
be rotatioanally unstable according to this
criteria. So how can they be turbulent
have anomalous viscosity?
Angular momentum has to be transferred outward to let the central star accrete the
matter of the disk, else the matter can’t approach the star.
Turbulence
• A collective phenomenon
• Unstable flows often evolve into a state
of motion called TURBULENCE, with a
chaotic 3-D vorticity field with broad
spectrum of small temporal and spatial
scales.
Leonardo da Vinci (1452-1519) wrote “Observe the motion
of the surface of the water, which resembles that of hair,
which has two motions, of which one is caused by the
weight of the hair, the other by the direction of the curls. Thus
the water has eddying motions, one part of which is due to the
principal current, the other to random and reverse motion.”
“... the smallest eddies are almost
numberless, and large
things are rotated only by large
eddies and not by small ones,
and small things are turned by
small eddies and large.”
Reynolds Experiment
• The configuration of
Reynolds experiment
on flow along a pipe.
• Experiment made
with 3 tubes of
different diameters.
• Streakes of colored
water entered the
tube with clear water.
From Reynolds 1883, fig. 13
General results of the Reynolds
Experiment
•
•
•
(a)When the velocities were sufficiently
low, the streak of color extended in a
beautiful straight line through the tube.
(b)As the velocity was increased by
small stages, at some point in the tube,
the color band would all at once mix up
with the surrounding water, and fill the
rest of the tube with a mass of colored
water. (Any increase in the velocity
caused the point of break down to
approach the trumpet.
(c) On viewing the tube by the light of
an electric spark, the mass of color
resolved itself into a mass of more or
less distinct curls, showing eddies.
From Reynolds 1883, Figs 3,4 & 5
Intermittency
From Reynolds 1883, Fig. 16.
• Just above the critical Reynolds number, turbulence reveals
its intermittent character:
• The disturbance would suddenly come on through a certain
length of the tube and pass away and then come on again,
giving the appereance of flashes and these flashes would
often comence successively at one point in pipe.
• Such flashes are now called “turbulent spots” or “turbulent
bursts.
Definition of Turbulence-1
T. von Karman qoutes G. I. Taylor with the
following definition of turbulence:
“Turbulence is an irregular motion which in
general makes its appearance in fluids,
gaseous or liquid, when they past
solid surfaces or even when neighboring
streams of the same fluid flow past or over
one another.”
Definition of Turbulence-2
• J.O. Hinze,(in his book onTurbulence),
offers yet another definition:
Turbulent fluid motion is an irregular
condition of the flow in which the various
quantities show a random variation with
time and space coordinates, so that
statistically distinct average values can be
discerned.”
A Modern Definition of Turbulence
• Turbulence is any chaotic solution to the
3-D Navier-Stokes equations that is
sensitive to initial data and which occurs
as a result of successive instabilities of
laminar flows as a bifurcation parameter is
increased through a succession of values.
Universally Accepted Features of
Turbulence
• Disorganized, chaotic, seemingly random behavior;
• No repeatability (i.e., sensitivity to initial conditions);
• Extremely large range of length and time scales (but
such that the smallest scales are still sufficiently large to
satisfy the continuum hypothesis);
• Enhanced diffusion (mixing) and dissipation (both of
which are mediated by viscosity at molecular scales);
• Three dimensionality, time dependence and rotationality
(hence, potential flow cannot be turbulent because it is
by definition irrotational);
• Intermittency in both space and time.
MHD Turbulence
• When the fluid is conducting turbulent motions are accompanied by
B-field fluctuations.
• Alfvén effect: small scale fluctuations are weakly interacting Alfvén
waves propagating along the large scale field (Iroshnikov 1964,
Kraichnan 1965).
• Energy spectrum predicted to be flatter (k-3/2) than the Kolmogorov
spectrum (k-5/3).
• BUT the energy spectrum of the Solar wind is closer to the
Kolmogorov spectrum
• Solution: MHD turbulence is unisotropic (Goldreich & Sridhar 1995)spectrum is more strongly developed perpendicular to the local Bfield where the Alfvén effect is not operative. This explains why
Kolmogorov-like dynamics is effective.
• Self-organization processes in MHD turbulence that have no
hydrodynamic counterpart.
Magnetorotational Instability
• In the presence of a weak magnetic field
the criterion for instability changes
profoundly and Keplerian disks are
unstable WRT to the new criteria.
The Great
Red Spot of
Jupiter
…has been
observed for
300 years although
the expected life time is
about 1 day
How could such a
large eddy manage
to persist over the
centuries?
Hook, R., “A spot in one of the belts of Jupiter” Phil. Trans. 1, 3 (1665)
• The turbulence in
Jupiter's
atmosphere does
not behave
similar to
homogenious,
isotropic
turbulence in
which vortices
randomly merge,
cascading their
energies into
progressively
smaller scales.
.GRS swallows only eddies of its own sign and repells the opposites
.This strange behavior depends critically on the dynamics in Jupiters
atmosphere being 2D in character
.The GRS is a nearly 2D vortex not attached to any topographic feature
Marcus, P. S., Jupiter's Great Red Spot and other vortices,
Annu. Rev. Astron. Astrophys. 31, 523-573 (1993)
References
links
• http://www.galleryoffluidmechanics.com/
• http://www.navier-stokes.net/
• http://www.fluidmech.net/
•
http://www.enseeiht.fr/hmf/travaux/CD0001/travaux/optmfn/hi/01pa/hyb72/index.htm
• http://perso.wanadoo.fr/laurent.nack/
• http://www.itsc.com/movies/index.htm
• http://woodall.ncsa.uiuc.edu/dbock/projects/3drt/
• http://astron.berkeley.edu/~jrg/ay202/lectures.html
Other References
• “Waves in Fluids”, Lighthill (1978)
• “Hydrodynamic stability” Drazin and Reid
(1981)